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Primary Maths · Year 4Chapter 9 · Shapes, angles and symmetry
Year 4 · Chapter 9

Year 4 Shapes, Angles and Symmetry: worked examples and questions with answers

By the end of this chapter you will be able to

  • Describe turns as quarter, half, three-quarter and whole turns, and know that a right angle is a quarter turn
  • Identify acute, right and obtuse angles and a straight line, and compare and order angles up to two right angles
  • Name triangles (equilateral, isosceles, scalene and right-angled) and say how you know
  • Name quadrilaterals (square, rectangle, rhombus, parallelogram, trapezium and kite) and describe them using sides, angles and parallel and perpendicular lines
  • Name regular and irregular polygons, such as pentagons, hexagons, heptagons and octagons
  • Sort shapes using Venn and Carroll diagrams, and solve “Which shape am I?” riddles
  • Find lines of symmetry in shapes, letters and patterns, whichever way round they are
  • Complete a symmetric figure on a grid using a vertical, horizontal or diagonal mirror line

You should already know

  • Names of 2-D shapes: square, rectangle, triangle, circle (Years 1 to 3)
  • A right angle is a quarter turn (Year 3)
  • Counting sides and corners
  • Fractions of a turn: quarter, half and three-quarter (Chapter 5)
  • Telling the time on a clock face (Chapter 8)

Where this leads

  • Position and direction: coordinates and translations (Chapter 10)
  • Measuring angles with a protractor and angles in degrees (Year 5)
  • Reflection and translation of shapes (Year 5)
  • Angles on a line and at a point, and angles in shapes (Years 5 and 6)

National Curriculum for England, Key Stage 2 mathematics, Year 4. Content: Compare and classify geometric shapes, including quadrilaterals and triangles, based on their properties and sizes, Identify acute and obtuse angles and compare and order angles up to two right angles by size, Identify lines of symmetry in 2-D shapes presented in different orientations, Complete a simple symmetric figure with respect to a specific line of symmetry.

Printable worksheets. Every lesson in this chapter has a free worksheet and an answer sheet to download and print.

Exercise P  Do you remember?

Non-calculator
1
FWhat is the name of this shape?
2
FHow many right angles make a whole turn?
3
FWhich pair of lines is parallel? Write the letter.
ABC
4
FThe minute hand of a clock moves from the 12 to the 3. What turn is this?
12369
5
FWhat is the name of this shape? It has four equal sides and four right angles.
9.1

Turns and right angles

Year 4
Angles as turns, Right angles, Perpendicular and parallel lines

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Turns and right anglesComing soon

When you spin round on the spot, you make a turn. A whole turn takes you all the way round until you face the way you started. A half turn takes you half the way round, so you face the opposite way. A quarter turn is one quarter of the way round, like turning to look at the door beside you. A three-quarter turn is three quarters of the way round.

startendquarter turnstartendhalf turnstartendthree-quarter turnstartwhole turn
The pointer starts facing up. The orange arrow shows the turn. All of these turns go clockwise.

An angle is the amount of turn between two lines that meet at a point. The corner of a book, a door or a window makes a very special angle. It is called a right angle. A right angle is a quarter turn. We mark a right angle with a small square.

a right anglethe corner of a book
A right angle looks like the corner of a piece of paper.

A whole turn is made from 4 right angles. So a half turn is 2 right angles, and two right angles together make a straight line. A three-quarter turn is 3 right angles.

TurnFraction of a whole turnRight angles
quarter turn141
half turn122 (a straight line)
three-quarter turn343
whole turn14

Clockwise and anticlockwise. Clockwise is the way the hands of a clock go. Anticlockwise is the other way. On a compass, going clockwise from North you reach East, then South, then West, then North again. Each step is a quarter turn.

startendclockwisestartendanticlockwiseNESWN, E, S, W
The same quarter turn can go clockwise or anticlockwise. It ends up in a different place.

Perpendicular and parallel lines. Two lines that meet at a right angle are perpendicular. Two lines that stay the same distance apart and never meet are parallel.

perpendicularparallel
Perpendicular lines make a right angle. Parallel lines never meet.

Key ideas

  • A whole turn = 4 right angles. A half turn = 2 right angles. A quarter turn = 1 right angle. A three-quarter turn = 3 right angles.
  • A right angle is a quarter turn. It looks like the corner of a book. It is marked with a small square.
  • Two right angles make a straight line.
  • Clockwise goes the way the hands of a clock go. Anticlockwise goes the other way.
  • Lines that meet at a right angle are perpendicular. Lines that never meet are parallel.
Worked example 9.1.1
The minute hand of a clock moves clockwise from the 12 to the 6. What turn is this? Write the number of right angles.
12369

Key words. A right angle is a quarter turn, like the corner of a book. Two right angles make a half turn (a straight line). Three make a three-quarter turn. Four make a whole turn. Clockwise is the way the hands of a clock go. Anticlockwise is the other way.

Method. The numbers 12, 3, 6 and 9 on a clock are a quarter turn apart. Count the quarter turns the minute hand makes as it moves clockwise.

  1. Find the start and the end. We look at where the hand starts and where it stops.
    It starts at 12 and moves clockwise to 6.
  2. Count the quarter turns. Each jump from one of 12, 3, 6, 9 to the next is a quarter turn.
    12 → 3 → 6
    2 quarter turns
  3. Name the turn. 2 out of 4 quarters.
    It is a half turn (2 right angles).

Check. A whole turn takes 60 minutes. A half turn takes 30 minutes. ✓

Watch out. The minute hand always moves clockwise.

Worked example 9.1.2
Mia is facing East. She makes a quarter turn anticlockwise. Which way is she facing now?
NESW

Key words. A right angle is a quarter turn, like the corner of a book. Two right angles make a half turn (a straight line). Three make a three-quarter turn. Four make a whole turn. Clockwise is the way the hands of a clock go. Anticlockwise is the other way.

Method. Turn one step at a time. Remember the order of the compass points going clockwise: North, East, South, West. A quarter turn clockwise moves one place along.

  1. Remember the order of the compass points going clockwise. A quarter turn clockwise moves one place along: N, E, S, W. Anticlockwise goes the other way.
    North → East → South → West → North
  2. Turn a quarter turn anticlockwise. 1 quarter turn. Clockwise goes N → E → S → W → N. Anticlockwise goes the other way.
    Start facing East.
    East → North
  3. Write the answer.
    The person is now facing North.
    NESW

Check. Going clockwise, North = 0, East = 1, South = 2, West = 3 quarter turns from North. The total of the turns gives the same direction. ✓

Watch out. Clockwise: N → E → S → W. Anticlockwise is the opposite way, so N → W → S → E.

Worked example 9.1.3
Is this angle a right angle? Use the corner of a piece of paper.

Key words. An angle is the amount of turn between two lines that meet. An acute angle is smaller than a right angle. An obtuse angle is bigger than a right angle but smaller than a straight line. A straight line is two right angles.

Method. Compare the angle with the corner of a piece of paper. The corner of paper is a right angle.

  1. Put the corner of a piece of paper on the angle. Line up one side of the paper with one arm of the angle. The dashed lines show the paper corner.
  2. Look at the other arm. It tells us if the angle is the same as the paper corner, smaller, or bigger.
    The other arm lines up exactly with the other side of the paper corner.
  3. Answer.
    It is a right angle.

Check. A right angle is a quarter turn. ✓

Watch out. Look at the size of the opening, not the length of the arms.

Worked example 9.1.4
Which pair of lines is perpendicular? Write the letter.
ABC

Key words. A quadrilateral has 4 straight sides. Parallel lines stay the same distance apart and never meet. Perpendicular lines meet at a right angle. A trapezium has exactly one pair of parallel sides.

Method. Check each pair of lines. Perpendicular lines are lines that meet at a right angle.

  1. Remember what we are looking for. We want the pair that is perpendicular.
    Perpendicular lines meet at a right angle (a square corner).
  2. Look at each pair of lines. Decide if each pair is parallel, perpendicular or neither.
    A: parallel
    B: perpendicular
    C: neither
  3. Choose the pair. Only pair B meets at a right angle.
    The perpendicular pair is B.

Check. Pair B is the only one that is perpendicular. ✓

Watch out. Lines that cross at an angle that is not a right angle are neither.

Worked example 9.1.5
How many right angles are in a three-quarter turn?

Key words. A right angle is a quarter turn, like the corner of a book. Two right angles make a half turn (a straight line). Three make a three-quarter turn. Four make a whole turn. Clockwise is the way the hands of a clock go. Anticlockwise is the other way.

Method. A right angle is a quarter turn. A whole turn is 4 quarter turns. Count how many quarter turns make the turn you are asked about.

  1. Think about the whole turn. A whole turn is all the way round. It is made of 4 right angles.
    1 whole turn = 4 right angles
  2. Find how many quarters are in a three-quarter turn. A three-quarter turn is 3 out of the 4 quarter turns.
    three-quarter turn = 3 quarter turns
  3. Write the answer. Each quarter turn is one right angle.
    three-quarter turn = 3 right angles

Check. 3 out of 4 quarters is 34 of a whole turn. ✓

Watch out. A half turn is 2 right angles (a straight line), not 1.

Worked example 9.1.6
The pointer turns as shown. Name the turn and say how many right angles it makes.
startend

Key words. A right angle is a quarter turn, like the corner of a book. Two right angles make a half turn (a straight line). Three make a three-quarter turn. Four make a whole turn. Clockwise is the way the hands of a clock go. Anticlockwise is the other way.

Method. Follow the arrow. Count how many quarter turns the pointer makes. A quarter turn is a right angle.

  1. Find where the pointer starts and where it ends. We say which way it points: up, right, down or left.
    Start: down.
    End: left.
  2. Follow the arrow and count the quarter turns. The arrow goes anticlockwise. Each time the pointer moves to the next of up, right, down and left, that is one quarter turn.
    down → right → up → left
    That is 3 quarter turns.
  3. Name the turn. 3 out of 4 quarters.
    It is a three-quarter turn anticlockwise.
  4. Write the number of right angles. Each quarter turn is one right angle.
    3 right angles

Check. 3 quarters out of 4 is 34 of a whole turn. ✓

Watch out. Count the quarter turns the arrow makes, not the number of directions you list.

Watch out

A half turn is 2 right angles, not 1. A right angle is a quarter turn. Two of them make a half turn (a straight line). Also, a clockwise turn and an anticlockwise turn of the same size end in different places, unless it is a half turn or a whole turn.

Exercise 9A  Turns, right angles and perpendicular lines

Non-calculator
1
FHow many right angles are there in each turn?
(a)a quarter turn
(b)a half turn
(c)a three-quarter turn
(d)a whole turn
2
FName each turn. Write quarter, half, three-quarter or whole turn.
(a)startend
(b)startend
(c)startend
(d)start
3
FEach person turns. Which way are they facing at the end? Write North, East, South or West.
(a)NESW Facing North. Turn a quarter turn clockwise.
(b)NESW Facing South. Turn a half turn clockwise.
(c)NESW Facing West. Turn a quarter turn anticlockwise.
(d)NESW Facing East. Turn a three-quarter turn clockwise.
4
FThe minute hand of a clock moves clockwise. Name each turn.
(a)12369   from the 12 to the 3
(b)12369   from the 3 to the 9
(c)12369   from the 9 to the 12
(d)12369   from the 6 to the 6
5
FIs each angle a right angle? Write yes or no. You can use the corner of a piece of paper.
(a)
(b)
(c)
(d)
6
FWhich pair of lines is perpendicular? Write the letter.
ABC
7
FWhich pair of lines is parallel? Write the letter.
ABC
8
FWrite parallel, perpendicular or neither for each pair of lines.
(a)
(b)
(c)
9
FHow many right angles does each shape have?
(a)
(b)
(c)
10
PSA robot is facing North. It makes a quarter turn clockwise, then a half turn, then a quarter turn anticlockwise. Which way is the robot facing now?
NESW
11
PSAisha turns through 3 right angles. (a) How many more right angles does she need to make a whole turn? (b) What fraction of a whole turn has she made?
12
RTom says, “A half turn is 1 right angle.” Is Tom correct? Explain how you know.
13
RWrite always, sometimes or never for each sentence.
(a)Two right angles make a straight line.
(b)A turn clockwise ends facing the same way as a turn of the same size anticlockwise.
(c)Perpendicular lines are parallel.
14
RLeo faces West. He makes a three-quarter turn clockwise, and then a quarter turn clockwise. (a) Which way does he face at the end? (b) What is the total turn he made?
NESW
9.2

Acute and obtuse angles

Year 4
Acute and obtuse angles, Comparing and ordering angles, Angles in shapes

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Acute and obtuse anglesComing soon

We use a right angle to describe other angles. The corner of a piece of paper is a right angle, so you can use it as an angle tester: put it on one arm of an angle and see where the other arm goes.

acuteright angleobtusestraight line
The dashed lines are the corner of a piece of paper. An acute angle fits inside it. An obtuse angle is wider.

Comparing angles. To compare two angles, look at how far apart the two arms are. The wider the opening, the bigger the angle. The length of the arms does not matter. You can draw an angle with very long arms, and it can still be small.

AA is smallerBB is bigger
Angle A has long arms but opens less. Angle B has short arms but opens more. Angle B is the bigger angle.

Ordering angles. To put angles in order, first sort them: acute angles are the smallest, then a right angle, then obtuse angles. Then compare angles in the same group by how wide they open.

Angles in shapes. Every corner of a shape is an angle. You can use the angle tester on each corner. A triangle can have acute angles, a right angle or one obtuse angle.

ABCtwo acute angles, one obtusetwo acute, two obtuse
Orange arcs mark the angles. A small square would mark a right angle.

Key ideas

  • Acute: smaller than a right angle. Right angle: a quarter turn. Obtuse: bigger than a right angle, smaller than a straight line.
  • A straight line is an angle of two right angles.
  • Use the corner of a piece of paper as an angle tester.
  • Compare angles by how wide they open, not by how long the arms are.
  • To order angles: acute first, then right angle, then obtuse. Then compare inside each group.
Worked example 9.2.1
Is this angle acute, a right angle or obtuse?

Key words. An angle is the amount of turn between two lines that meet. An acute angle is smaller than a right angle. An obtuse angle is bigger than a right angle but smaller than a straight line. A straight line is two right angles.

Method. Compare the angle with a right angle (the corner of a piece of paper). Smaller is acute. The same is a right angle. Bigger, but less than a straight line, is obtuse.

  1. Put the paper corner on one arm of the angle. The dashed lines show the corner of the paper.
  2. Look at the other arm. Is it inside the corner, on the corner, or past the corner?
    The other arm is inside the paper corner. The angle is smaller than a right angle.
  3. Name the angle.
    It is acute.

Check. Acute < right angle < obtuse < straight line. ✓

Watch out. Look at how wide the angle opens, not how long the arms are.

Worked example 9.2.2
Is this angle acute, a right angle or obtuse?

Key words. An angle is the amount of turn between two lines that meet. An acute angle is smaller than a right angle. An obtuse angle is bigger than a right angle but smaller than a straight line. A straight line is two right angles.

Method. Compare the angle with a right angle (the corner of a piece of paper). Smaller is acute. The same is a right angle. Bigger, but less than a straight line, is obtuse.

  1. Put the paper corner on one arm of the angle. The dashed lines show the corner of the paper.
  2. Look at the other arm. Is it inside the corner, on the corner, or past the corner?
    The other arm is outside the paper corner. It is bigger than a right angle, but it is smaller than a straight line.
  3. Name the angle.
    It is obtuse.

Check. Acute < right angle < obtuse < straight line. ✓

Watch out. Look at how wide the angle opens, not how long the arms are.

Worked example 9.2.3
What is the name of this angle?

Key words. An angle is the amount of turn between two lines that meet. An acute angle is smaller than a right angle. An obtuse angle is bigger than a right angle but smaller than a straight line. A straight line is two right angles.

Method. Compare the angle with a right angle (the corner of a piece of paper). Smaller is acute. The same is a right angle. Bigger, but less than a straight line, is obtuse.

  1. Put the paper corner on one arm of the angle. The dashed lines show the corner of the paper.
  2. Look at the other arm. Is it inside the corner, on the corner, or past the corner?
    The two arms make one straight line. That is two right angles.
  3. Name the angle.
    It is a straight line (two right angles).

Check. Acute < right angle < obtuse < straight line. ✓

Watch out. Look at how wide the angle opens, not how long the arms are.

Worked example 9.2.4
Which angle is bigger, A or B? Write the letter.
AB

Key words. An angle is the amount of turn between two lines that meet. An acute angle is smaller than a right angle. An obtuse angle is bigger than a right angle but smaller than a straight line. A straight line is two right angles.

Method. Compare how wide each angle opens. Use the corner of a piece of paper to help. The biggest angle opens the widest.

  1. Test each angle with the paper corner. The dashed lines show a right angle on each one.
    AB
  2. Sort them. Acute is smaller than a right angle. Obtuse is bigger.
    A: acute
    B: acute
  3. Find the biggest angle. Look at the group that has the biggest angles, then compare the openings.
    The biggest angle is B.

Check. No other angle opens wider than B. ✓

Watch out. Do not choose the angle with the longest arms. Look at the opening.

Worked example 9.2.5
Put these angles in order, from the smallest to the biggest.
ABCD

Key words. An angle is the amount of turn between two lines that meet. An acute angle is smaller than a right angle. An obtuse angle is bigger than a right angle but smaller than a straight line. A straight line is two right angles.

Method. Sort the angles into acute, right and obtuse using the corner of a piece of paper. Acute angles are the smallest. Then compare the angles that are in the same group.

  1. Test each angle with the paper corner. The dashed lines show the paper corner on each angle.
    ABCD
  2. Sort them into groups. Acute angles are smaller than a right angle. Obtuse angles are bigger than a right angle.
    B and D: acute (smaller than a right angle)
    A and C: obtuse (bigger than a right angle)
  3. Compare the angles in the same group. The wider the opening, the bigger the angle. Ignore how long the arms are.
    B < D < C < A
  4. Write the order. Start with the smallest.
    B, D, C, A

Check. Look at the first and last angle: B is the smallest and A is the biggest. ✓

Watch out. A long arm does not make an angle bigger. Only the size of the opening matters.

Worked example 9.2.6
Triangle ABC is shown. Which angle is obtuse?
ABC

Key words. An angle is the amount of turn between two lines that meet. An acute angle is smaller than a right angle. An obtuse angle is bigger than a right angle but smaller than a straight line. A straight line is two right angles.

Method. Compare each angle of the shape with a right angle. The obtuse angle is the one that is bigger than a right angle.

  1. Look at each angle. The orange marks show the angles. A small square means a right angle.
    ABC
  2. Test each angle with the paper corner. Smaller than the paper corner is acute. Bigger is obtuse.
    Angle A: obtuse
    Angle B: acute
    Angle C: acute
  3. Choose the angle.
    The obtuse angle is at A.

Check. Angle A really is obtuse. ✓

Watch out. Check every corner.

Watch out

Long arms do not make a bigger angle. Angle size is the amount of turn between the arms. Two angles can have arms of different lengths and still be the same size. Also take care with the words: acute is the small angle and obtuse is the wide one.

Exercise 9B  Acute, right and obtuse angles

Non-calculator
1
FName each angle. Write acute, right angle or obtuse.
(a)
(b)
(c)
(d)
2
FIs each angle acute or obtuse?
(a)
(b)
(c)
(d)
3
FWrite the missing word: acute, obtuse, right angle or straight line.
(a)An angle that is smaller than a right angle is called ___.
(b)An angle that is bigger than a right angle but smaller than a straight line is ___.
(c)Two right angles together make a ___.
(d)A quarter turn is a ___.
4
FWhich angle is the biggest? Write the letter.
ABCD
5
FWhich angle is the smallest? Write the letter.
ABCD
6
FWrite the angles in order, smallest first.
ABCD
7
FThe clock shows an o’clock time. What kind of angle do the hands make? Write acute, right angle, obtuse or straight line.
(a)12369   2 o’clock
(b)12369   3 o’clock
(c)12369   5 o’clock
(d)12369   6 o’clock
8
FWhich angle in triangle ABC is obtuse? Write the letter.
ABC
9
FThis shape has four angles.
(a)How many acute angles does the shape have?
(b)How many obtuse angles does the shape have?
10
PSEvery angle in a regular polygon is the same size. How many obtuse angles does each regular polygon have?
(a)   regular hexagon
(b)   regular octagon
11
PSMei draws 9 angles. 4 are acute and 2 are right angles. The rest are obtuse. How many obtuse angles does Mei draw?
12
RLeo says, “Angle A is bigger than angle B because its arms are longer.” Is Leo right? Explain.
AB
13
RWrite always, sometimes or never for each sentence.
(a)An obtuse angle is bigger than a right angle.
(b)Two acute angles put together make an obtuse angle.
(c)A right angle is an acute angle.
14
RZainab wants to draw two obtuse angles side by side on a straight line. Can she do it? Explain your answer using right angles.
9.3

Triangles

Year 4
Triangles: equilateral, isosceles, scalene, right-angled, Properties of triangles

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: TrianglesComing soon

A triangle is a shape with 3 straight sides, 3 corners (also called vertices) and 3 angles. We name triangles by looking at their sides and their angles.

equilateralisosceles7 cm5 cm4 cmscaleneright-angled
Little tick marks show sides that are the same length. Scalene has no ticks. The small square shows a right angle.

Tick marks. In a picture, sides with the same number of tick marks are equal. A right-angled triangle can be scalene (like the one above, with sides 3, 4 and 5) or isosceles (if two sides next to the right angle are equal).

A triangle looks different when you turn it round, but it is still the same triangle. An isosceles triangle can point up, down or sideways.

These are all the same isosceles triangle, turned round.

Key ideas

  • Equilateral: 3 equal sides. Isosceles: 2 equal sides. Scalene: no equal sides.
  • Right-angled: has one right angle (shown by a small square).
  • Tick marks show sides of equal length.
  • An isosceles triangle has two equal angles. An equilateral triangle has three equal angles.
  • Turning a triangle round does not change its name.
Worked example 9.3.1
What kind of triangle is this? Write equilateral, isosceles or scalene.

Key words. A triangle has 3 straight sides. Equilateral: 3 equal sides. Isosceles: 2 equal sides. Scalene: no equal sides. Right-angled: one right angle. Little tick marks show sides that are equal.

Method. Count how many sides are equal. Three equal sides: equilateral. Two equal sides: isosceles. No equal sides: scalene. A right angle makes it right-angled too.

  1. Look at the three sides of the triangle. Tick marks show sides that are equal.
    All three sides have one tick mark, so all three sides are equal.
  2. Count the equal sides. The number of equal sides tells us the name.
    There are 3 equal sides.
  3. Name the triangle. Three equal sides is equilateral. Two equal sides is isosceles. No equal sides is scalene.
    The triangle has 3 equal sides: equilateral.

Check. Equilateral = 3 equal, isosceles = 2 equal, scalene = 0 equal. This one has 3. ✓

Watch out. Isosceles has exactly two equal sides in this book. Equilateral has all three.

Worked example 9.3.2
What kind of triangle is this? The side lengths are shown.
8 cm5 cm5 cm

Key words. A triangle has 3 straight sides. Equilateral: 3 equal sides. Isosceles: 2 equal sides. Scalene: no equal sides. Right-angled: one right angle. Little tick marks show sides that are equal.

Method. Count how many sides are equal. Three equal sides: equilateral. Two equal sides: isosceles. No equal sides: scalene. A right angle makes it right-angled too.

  1. Look at the three sides of the triangle. Sides that have the same length are equal.
    8 cm5 cm5 cm
    The sides are 8 cm, 5 cm and 5 cm.
  2. Count the equal sides. The number of equal sides tells us the name.
    There are 2 equal sides.
  3. Name the triangle. Three equal sides is equilateral. Two equal sides is isosceles. No equal sides is scalene.
    The triangle has 2 equal sides: isosceles.

Check. Equilateral = 3 equal, isosceles = 2 equal, scalene = 0 equal. This one has 2. ✓

Watch out. Isosceles has exactly two equal sides in this book. Equilateral has all three.

Worked example 9.3.3
Triangle with sides 7 cm, 5 cm and 4 cm. What kind of triangle is it?
7 cm5 cm4 cm

Key words. A triangle has 3 straight sides. Equilateral: 3 equal sides. Isosceles: 2 equal sides. Scalene: no equal sides. Right-angled: one right angle. Little tick marks show sides that are equal.

Method. Count how many sides are equal. Three equal sides: equilateral. Two equal sides: isosceles. No equal sides: scalene. A right angle makes it right-angled too.

  1. Look at the three sides of the triangle. Sides that have the same length are equal.
    7 cm5 cm4 cm
    The sides are 7 cm, 5 cm and 4 cm.
  2. Count the equal sides. The number of equal sides tells us the name.
    No two sides are equal.
  3. Name the triangle. Three equal sides is equilateral. Two equal sides is isosceles. No equal sides is scalene.
    The triangle has no equal sides: scalene.

Check. Equilateral = 3 equal, isosceles = 2 equal, scalene = 0 equal. This one has 0. ✓

Watch out. Isosceles has exactly two equal sides in this book. Equilateral has all three.

Worked example 9.3.4
This triangle has sides 4 cm, 3 cm and 5 cm and a right angle. Give it its full name.
4 cm5 cm3 cm

Key words. A triangle has 3 straight sides. Equilateral: 3 equal sides. Isosceles: 2 equal sides. Scalene: no equal sides. Right-angled: one right angle. Little tick marks show sides that are equal.

Method. Count how many sides are equal. Three equal sides: equilateral. Two equal sides: isosceles. No equal sides: scalene. A right angle makes it right-angled too.

  1. Look at the three sides of the triangle. Sides that have the same length are equal.
    4 cm5 cm3 cm
    The sides are 4 cm, 5 cm and 3 cm.
  2. Count the equal sides. The number of equal sides tells us the name.
    No two sides are equal.
  3. Name the triangle. Three equal sides is equilateral. Two equal sides is isosceles. No equal sides is scalene.
    The triangle has no equal sides: scalene.
  4. Look for a right angle. A triangle with a right angle is also called right-angled. The little square marks it.
    There is a right angle, so it is a right-angled triangle.

Check. Equilateral = 3 equal, isosceles = 2 equal, scalene = 0 equal. This one has 0. ✓

Watch out. Isosceles has exactly two equal sides in this book. Equilateral has all three.

Worked example 9.3.5
Give the full name of this triangle. It has two sides the same and a right angle.

Key words. A triangle has 3 straight sides. Equilateral: 3 equal sides. Isosceles: 2 equal sides. Scalene: no equal sides. Right-angled: one right angle. Little tick marks show sides that are equal.

Method. Count how many sides are equal. Three equal sides: equilateral. Two equal sides: isosceles. No equal sides: scalene. A right angle makes it right-angled too.

  1. Look at the three sides of the triangle. Tick marks show sides that are equal.
    Two sides have the same tick mark, so two sides are equal. The third side has no tick.
  2. Count the equal sides. The number of equal sides tells us the name.
    There are 2 equal sides.
  3. Name the triangle. Three equal sides is equilateral. Two equal sides is isosceles. No equal sides is scalene.
    The triangle has 2 equal sides: isosceles.
  4. Look for a right angle. A triangle with a right angle is also called right-angled. The little square marks it.
    There is a right angle, so it is a right-angled isosceles triangle.

Check. Equilateral = 3 equal, isosceles = 2 equal, scalene = 0 equal. This one has 2. ✓

Watch out. Isosceles has exactly two equal sides in this book. Equilateral has all three.

Worked example 9.3.6
Which angle in this triangle is the right angle? Write the letter.
ABC

Key words. A triangle has 3 straight sides. Equilateral: 3 equal sides. Isosceles: 2 equal sides. Scalene: no equal sides. Right-angled: one right angle. Little tick marks show sides that are equal.

Method. Test each angle of the triangle with the corner of a piece of paper. The right angle fits the corner exactly.

  1. Look at the three angles. The little square shows where the right angle is.
    ABC
  2. Test each angle. Acute is smaller than the paper corner. Obtuse is bigger.
    Angle A: right angle
    Angle B: acute
    Angle C: acute
  3. Answer.
    The right angle is at A.

Check. A triangle can have only one right angle. ✓

Watch out. The right angle is not always at the bottom left.

Worked example 9.3.7
Triangle ABC is isosceles. Which two angles are equal?
ABC

Key words. A triangle has 3 straight sides. Equilateral: 3 equal sides. Isosceles: 2 equal sides. Scalene: no equal sides. Right-angled: one right angle. Little tick marks show sides that are equal.

Method. In an isosceles triangle two sides are equal, and two angles are equal. The equal angles are at the ends of the third side, opposite the equal sides.

  1. Find the two equal sides. The tick marks show them.
    ABC
    Sides AB and AC are equal.
  2. Find the angle where the equal sides meet. This is the top of the triangle (the apex). It is on its own.
    The equal sides meet at A.
  3. Name the two equal angles. The other two angles are the same size as each other.
    The equal angles are at B and C.

Check. Angle B = 53° and angle C = 53° in the drawing: they are the same size. ✓

Watch out. The equal angles are not at the point where the equal sides meet.

Watch out

Do not trust the way the triangle is turned. An isosceles triangle upside down, or lying on its side, is still isosceles. Look at the sides (tick marks or lengths), not at which way the triangle points. Also, a triangle can have only one right angle.

Exercise 9C  Triangles

Non-calculator
1
FName each triangle from the lengths of its sides (in cm). Write equilateral, isosceles or scalene.
(a)4 cm4 cm4 cm
(b)7 cm4 cm7 cm
(c)6 cm8 cm9 cm
(d)5 cm5 cm3 cm
2
FUse the tick marks to name each triangle.
(a)
(b)
(c)
(d)6 cm5 cm7 cm
3
FWhich triangle is right-angled? Write the letter.
ABC
4
FWrite how many equal sides each triangle has.
(a)an equilateral triangle
(b)an isosceles triangle
(c)a scalene triangle
5
FTriangle PQR is drawn. (a) Which two sides are equal? (b) What kind of triangle is it?
PQR7 cm5 cm5 cm
(a)Which two sides are equal?
(b)What kind of triangle is it?
6
FEach triangle is isosceles. Which two angles are equal? Write the letters.
(a)ABC
(b)ABC
7
FWhich angle in the triangle is the right angle? Write the letter.
ABC
8
FLook at these six triangles. All the sides that are equal have tick marks. How many of the triangles are…
ABCDEF
(a)equilateral?
(b)isosceles?
(c)scalene?
9
PSAn equilateral triangle has a perimeter of 24 cm. How long is each side?
10
PSTom draws a triangle. It has two equal sides and one right angle. What names can Tom give his triangle?
11
RWhich triangle is the odd one out? Explain your answer.
ABC7 cm5 cm4 cmD
12
RWrite always, sometimes or never for each sentence.
(a)A triangle has three straight sides.
(b)A scalene triangle has a right angle.
(c)A triangle has two right angles.
13
RA and B are two corners of a triangle. The third corner is one of the points C, D, E or F on the dot grid.
ABCDEF
(a)Which points make an isosceles triangle? Write the letters.
(b)Which points make a right-angled triangle? Write the letters.
9.4

Quadrilaterals and parallel lines

Year 4
Quadrilaterals, Parallel and perpendicular lines, Properties of shapes

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Quadrilaterals and parallel linesComing soon

A quadrilateral is a shape with 4 straight sides and 4 corners. There are six special quadrilaterals to learn. To tell them apart we look at three things: the sides (which are equal?), the angles (any right angles?) and the parallel sides.

Parallel lines stay the same distance apart and never meet. We mark them with arrows (>). Perpendicular lines meet at a right angle.

ABCDAB and DC are parallel (same arrows)PQRSPQ and QR are perpendicular
Sides with the same arrow marks are parallel. A small square shows perpendicular sides.

Here are the six quadrilaterals and their properties. In this book a trapezium has exactly one pair of parallel sides.

ShapeNameEqual sidesRight anglesPairs of parallel sidesLines of symmetry
squareall 4 sides equal424
rectangleopposite sides equal422
rhombusall 4 sides equal022
parallelogramopposite sides equal020
trapeziumno equal sides010
kitetwo pairs of equal sides, next to each other001

Look carefully at the table. A square has everything: 4 equal sides, 4 right angles and 2 pairs of parallel sides. That means a square is a special kind of rectangle, and a special kind of rhombus. A rectangle has 4 right angles but only opposite sides are equal. A rhombus is like a leaning square: all sides equal but no right angles. A parallelogram is a leaning rectangle. A kite has two pairs of equal sides next to each other.

Key ideas

  • A quadrilateral has 4 straight sides and 4 corners.
  • Parallel sides never meet (arrows >). Perpendicular sides meet at a right angle (small square).
  • Square: 4 equal sides, 4 right angles. Rectangle: 4 right angles, opposite sides equal. Rhombus: 4 equal sides, no right angles.
  • Parallelogram: 2 pairs of parallel sides. Trapezium: 1 pair of parallel sides. Kite: 2 pairs of equal sides next to each other.
  • A square is a special rectangle. Turning a shape does not change its name.
Worked example 9.4.1
What is the name of this quadrilateral? Tick marks show equal sides and arrows show parallel sides.

Key words. A quadrilateral has 4 straight sides. Parallel lines stay the same distance apart and never meet. Perpendicular lines meet at a right angle. A trapezium has exactly one pair of parallel sides.

Method. Describe the shape: sides, then parallel sides, then right angles. Compare with the six quadrilaterals: square, rectangle, rhombus, parallelogram, trapezium and kite.

  1. Count the sides. A shape with 4 straight sides is a quadrilateral.
    It has 4 sides.
  2. Look at the sides. Tick marks show equal sides.
    The shape has all 4 sides equal.
  3. Look for parallel sides and right angles. Arrows on two sides mean they are parallel. A small square means a right angle.
    Pairs of parallel sides: 2.
    Right angles: 0.
  4. Decide on the name. All four sides are equal, but the angles are not right angles.
    It is a rhombus.

Check. The rhombus has all 4 sides equal, 2 pair(s) of parallel sides and 0 right angle(s). ✓

Watch out. Turning a shape round does not change its name.

Worked example 9.4.2
What is the name of this quadrilateral?

Key words. A quadrilateral has 4 straight sides. Parallel lines stay the same distance apart and never meet. Perpendicular lines meet at a right angle. A trapezium has exactly one pair of parallel sides.

Method. Describe the shape: sides, then parallel sides, then right angles. Compare with the six quadrilaterals: square, rectangle, rhombus, parallelogram, trapezium and kite.

  1. Count the sides. A shape with 4 straight sides is a quadrilateral.
    It has 4 sides.
  2. Look at the sides. Tick marks show equal sides.
    The shape has opposite sides equal.
  3. Look for parallel sides and right angles. Arrows on two sides mean they are parallel. A small square means a right angle.
    Pairs of parallel sides: 2.
    Right angles: 4.
  4. Decide on the name. Four right angles, and the sides are equal only in opposite pairs.
    It is a rectangle.

Check. The rectangle has opposite sides equal, 2 pair(s) of parallel sides and 4 right angle(s). ✓

Watch out. Turning a shape round does not change its name.

Worked example 9.4.3
What is the name of this quadrilateral?

Key words. A quadrilateral has 4 straight sides. Parallel lines stay the same distance apart and never meet. Perpendicular lines meet at a right angle. A trapezium has exactly one pair of parallel sides.

Method. Describe the shape: sides, then parallel sides, then right angles. Compare with the six quadrilaterals: square, rectangle, rhombus, parallelogram, trapezium and kite.

  1. Count the sides. A shape with 4 straight sides is a quadrilateral.
    It has 4 sides.
  2. Look at the sides. Tick marks show equal sides.
    The shape has no equal sides.
  3. Look for parallel sides and right angles. Arrows on two sides mean they are parallel. A small square means a right angle.
    Pairs of parallel sides: 1.
    Right angles: 0.
  4. Decide on the name. Exactly one pair of parallel sides.
    It is a trapezium.

Check. The trapezium has no equal sides, 1 pair(s) of parallel sides and 0 right angle(s). ✓

Watch out. Turning a shape round does not change its name.

Worked example 9.4.4
What is the name of this quadrilateral?

Key words. A quadrilateral has 4 straight sides. Parallel lines stay the same distance apart and never meet. Perpendicular lines meet at a right angle. A trapezium has exactly one pair of parallel sides.

Method. Describe the shape: sides, then parallel sides, then right angles. Compare with the six quadrilaterals: square, rectangle, rhombus, parallelogram, trapezium and kite.

  1. Count the sides. A shape with 4 straight sides is a quadrilateral.
    It has 4 sides.
  2. Look at the sides. Tick marks show equal sides.
    The shape has two pairs of equal sides, next to each other.
  3. Look for parallel sides and right angles. Arrows on two sides mean they are parallel. A small square means a right angle.
    Pairs of parallel sides: 0.
    Right angles: 0.
  4. Decide on the name. Two pairs of equal sides next to each other, and no parallel sides.
    It is a kite.

Check. The kite has two pairs of equal sides, next to each other, 0 pair(s) of parallel sides and 0 right angle(s). ✓

Watch out. Turning a shape round does not change its name.

Worked example 9.4.5
Name the pairs of parallel sides in parallelogram ABCD.
ABCD

Key words. A quadrilateral has 4 straight sides. Parallel lines stay the same distance apart and never meet. Perpendicular lines meet at a right angle. A trapezium has exactly one pair of parallel sides.

Method. Parallel sides stay the same distance apart and never meet. Arrow marks show sides that are parallel: sides with the same number of arrows are parallel to each other.

  1. Look at the arrow marks. Two sides with the same arrow marks are a pair of parallel sides.
    ABCD
  2. Write the sides with the same arrows. Use the letters at the two ends of each side.
    AB and CD; BC and DA
  3. Answer.
    The parallel sides are AB and CD; BC and DA.

Check. Pairs of parallel sides: 2. ✓

Watch out. Name each side with its two end letters, such as AB.

Worked example 9.4.6
Rectangle PQRS is drawn. Which sides are perpendicular to PQ?
PQRS

Key words. A quadrilateral has 4 straight sides. Parallel lines stay the same distance apart and never meet. Perpendicular lines meet at a right angle. A trapezium has exactly one pair of parallel sides.

Method. Perpendicular lines meet at a right angle. Look for the sides that meet PQ at a right angle.

  1. Look at the right angles. Small squares mark them.
    PQRS
  2. Find the sides that meet PQ at a right angle. These are the sides next to it.
    QR and SP
  3. Answer.
    QR and SP are perpendicular to PQ.

Check. Each of these meets PQ at a small square. ✓

Watch out. Sides opposite each other are parallel, not perpendicular.

Worked example 9.4.7
Is a square a rectangle? Explain.

Key words. A quadrilateral has 4 straight sides. Parallel lines stay the same distance apart and never meet. Perpendicular lines meet at a right angle. A trapezium has exactly one pair of parallel sides.

Method. A square has all the properties of a rectangle: 4 right angles and opposite sides equal and parallel. It also has all sides equal.

  1. List what a rectangle needs. A rectangle has 4 right angles, and its opposite sides are equal and parallel.
    rectangle: 4 right angles; opposite sides equal and parallel
  2. Check the square. A square has 4 right angles, and its opposite sides are equal and parallel.
    square: 4 right angles; opposite sides equal and parallel ✓
  3. Answer. The square has everything a rectangle needs.
    Yes, a square is a rectangle. It is a special rectangle with all four sides equal.

Check. A square is a rectangle, but most rectangles are not squares. ✓

Watch out. Not every rectangle is a square.

Watch out

A shape keeps its name when it is turned. A square standing on one corner is still a square (do not call it a diamond). Also, a rhombus is not a square: it has four equal sides but its corners are not right angles. And do not forget: only pairs of sides are parallel. A trapezium has exactly one pair.

Exercise 9D  Quadrilaterals

Non-calculator
1
FName each quadrilateral. Use the tick marks, arrows and squares to help.
(a)
(b)
(c)
(d)
(e)
(f)
2
FHow many right angles does each shape have?
(a)square
(b)rhombus
(c)kite
(d)rectangle
3
FHow many pairs of parallel sides does each shape have?
(a)square
(b)trapezium
(c)kite
(d)parallelogram
4
FName the pairs of parallel sides in this parallelogram.
ABCD
5
FRectangle PQRS is drawn. Which sides are perpendicular to PQ?
PQRS
6
FWhich quadrilateral am I? Write its name.
(a)I have 4 sides. All of my sides are the same length. I have four right angles.
(b)I have 4 sides. My opposite sides are equal in length. I have four right angles. My sides are not all the same length.
(c)I have 4 sides. I have exactly one pair of parallel sides.
7
FIs each statement true or false?
(a)A rectangle has four right angles.
(b)A rhombus has four right angles.
(c)A kite has two pairs of equal sides.
(d)A trapezium has two pairs of parallel sides.
8
FPriya says, “This shape is a square because all four sides are equal.” Is Priya right? Explain.
9
FWhich shape has two pairs of equal sides that are next to each other? Write the letter.
ABCD
10
PSA shape is cut from corner to corner into two triangles. What kind of triangles are they? Use the full name.
(a)A rectangle with sides 4 cm and 3 cm is cut along a diagonal.
(b)A square with sides 3 cm is cut along a diagonal.
11
RWrite always, sometimes or never for each sentence.
(a)A square is a rectangle.
(b)A rectangle is a square.
(c)A rhombus has four right angles.
(d)A trapezium has two pairs of parallel sides.
12
RWhich shape is the odd one out? Explain why.
ABCD
9.5

Polygons and sorting shapes

Year 4
Polygons: regular and irregular, Names of polygons, Sorting and classifying shapes (Venn and Carroll diagrams)

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Polygons and sorting shapesComing soon

A polygon is a flat shape that has straight sides and is closed (no gaps). A circle is not a polygon because it has a curved edge. The corners of a polygon are called vertices (one corner is a vertex). A polygon always has the same number of vertices as sides. We name a polygon by the number of its sides.

triangle (3)quadrilateral (4)pentagon (5)hexagon (6)heptagon (7)octagon (8)nonagon (9)decagon (10)
Regular polygons with 3 to 10 sides. The number in brackets is the number of sides.

Regular and irregular. A polygon is regular if all its sides are equal and all its angles are equal. If not, it is irregular. A square is a regular quadrilateral. A rectangle is irregular, because its sides are not all equal. A rhombus is irregular too, because its angles are not all equal.

regular hexagonirregular hexagonrectangle (irregular)rhombus (irregular)
Only the first shape has equal sides and equal angles.

Sorting shapes. We can sort shapes using a Carroll diagram (a table with yes and no) or a Venn diagram (circles). In a Venn diagram, a shape that fits both labels goes in the part where the circles overlap. A shape that fits neither goes outside the circles, but inside the box.

has a right angleno right angle
4 sidesA, BC, D
not 4 sidesEF
has 4 sideshas a right angleCDABEF
The same six shapes (A to F) sorted in a Carroll diagram and in a Venn diagram.

Key ideas

  • A polygon is a closed flat shape with straight sides. A vertex is a corner. Sides and vertices are the same number.
  • Names: triangle 3, quadrilateral 4, pentagon 5, hexagon 6, heptagon 7, octagon 8, nonagon 9, decagon 10.
  • Regular: all sides equal and all angles equal. Otherwise irregular.
  • Carroll diagram: a table with yes and no. Venn diagram: a shape with both labels goes in the overlap.
  • To solve “Which shape am I?”, use one clue at a time and cross out the shapes that do not fit.
Worked example 9.5.1
Write the name of this polygon.

Key words. A polygon is a flat shape with straight sides that is closed (no gaps). A vertex is a corner (the plural is vertices). A regular polygon has equal sides and equal angles. An irregular polygon does not.

Method. A polygon is named by its number of sides. Count the sides (or the corners, which are the same number) and use the name for that number.

  1. Count the sides. Number each corner as you count round the shape. The last number is the number of sides.
    1234567
    The shape has 7 sides.
  2. Remember the names of the polygons. The name depends on the number of sides.
    triangle 3, quadrilateral 4, pentagon 5, hexagon 6, heptagon 7, octagon 8, nonagon 9, decagon 10
  3. Find the name. Look for 7 in the list.
    7 sides: heptagon

Check. The shape has 7 corners as well as 7 sides. ✓

Watch out. Count carefully. Number each side so you do not count one twice.

Worked example 9.5.2
Write the full name of this polygon, including regular or irregular. All the sides are the same length and all the angles are the same size.

Key words. A polygon is a flat shape with straight sides that is closed (no gaps). A vertex is a corner (the plural is vertices). A regular polygon has equal sides and equal angles. An irregular polygon does not.

Method. A polygon is named by its number of sides. Count the sides (or the corners, which are the same number) and use the name for that number.

  1. Count the sides. Number each corner as you count round the shape. The last number is the number of sides.
    12345
    The shape has 5 sides.
  2. Remember the names of the polygons. The name depends on the number of sides.
    triangle 3, quadrilateral 4, pentagon 5, hexagon 6, heptagon 7, octagon 8, nonagon 9, decagon 10
  3. Find the name. Look for 5 in the list.
    5 sides: pentagon
  4. Is it regular? Regular means all sides equal and all angles equal.
    Yes, so it is a regular pentagon.

Check. The shape has 5 corners as well as 5 sides. ✓

Watch out. Count carefully. Number each side so you do not count one twice.

Worked example 9.5.3
Is this hexagon regular or irregular? All the sides are the same length. Explain.

Key words. A polygon is a flat shape with straight sides that is closed (no gaps). A vertex is a corner (the plural is vertices). A regular polygon has equal sides and equal angles. An irregular polygon does not.

Method. A regular polygon has all its sides equal and all its angles equal. Check both.

  1. Look at the sides. Tick marks show equal sides. Regular needs all sides equal.
    All the sides are equal.
  2. Look at the angles. Regular also needs all the angles equal. The orange arcs show the angles.
    The angles are not all equal.
  3. Decide. A regular polygon has equal sides and equal angles.
    This polygon is irregular.

Check. Regular needs both: sides equal, angles not equal. ✓

Watch out. Equal sides are not enough: a rhombus has four equal sides but it is not regular.

Worked example 9.5.4
Which of these shapes are polygons? Write the letters.
ABCD

Key words. A polygon is a flat shape with straight sides that is closed (no gaps). A vertex is a corner (the plural is vertices). A regular polygon has equal sides and equal angles. An irregular polygon does not.

Method. A polygon is a closed flat shape with only straight sides. Check each shape for two things: no gaps, and no curved sides.

  1. Remember what a polygon needs. It must be closed and have only straight sides.
    closed + straight sides = polygon
  2. Check each shape for gaps and curves. Look at every shape in turn.
    A: a closed shape with straight sides: a polygon.
    B: It has a curved edge and no straight sides, so it is not a polygon.
    C: It is not closed. There is a gap, so it is not a polygon.
    D: a closed shape with straight sides: a polygon.
  3. Pick the polygons.
    The polygons are A and D.

Check. Every polygon you picked is closed and has straight sides only. ✓

Watch out. A circle is not a polygon.

Worked example 9.5.5
Sort shapes A to F into the Carroll diagram. Write the letters in each box.
ABCDEF
has a right angleno right angle
4 sides
not 4 sides

Key words. A Venn diagram uses circles. Things that fit both labels go in the part where the circles overlap. A Carroll diagram is a table with yes and no.

Method. Test every shape against both labels. Then put it in the box that matches the answers: yes or no for each label.

  1. Look at each shape. Ask the two questions about each shape.
    A: yes for “has 4 sides”, yes for “has a right angle”
    B: yes for “has 4 sides”, yes for “has a right angle”
    C: yes for “has 4 sides”, no for “has a right angle”
    D: yes for “has 4 sides”, no for “has a right angle”
    E: no for “has 4 sides”, yes for “has a right angle”
    F: no for “has 4 sides”, no for “has a right angle”
  2. Fill in the Carroll diagram. Each shape goes in exactly one box.
    has a right anglenot: has a right angle
    has 4 sidesA, BC, D
    not: has 4 sidesEF
  3. Check. Count the shapes.
    6 shapes are in the diagram. There are 6 shapes. They all fit.

Check. Every shape is in exactly one box. ✓

Watch out. Every shape belongs in a box, even if it does not have either property.

Worked example 9.5.6
Sort shapes A to F into a Venn diagram with two circles. Which letters go in each part?
ABCDEF
has parallel sidesall sides equal

Key words. A Venn diagram uses circles. Things that fit both labels go in the part where the circles overlap. A Carroll diagram is a table with yes and no.

Method. Test each shape against both labels. Both yes: the middle where the circles overlap. Only the left label: the left circle only. Only the right label: the right circle only. Neither: outside the circles.

  1. Test each shape. Ask the two questions about each shape.
    A: “has parallel sides” yes, “all sides equal” no
    B: “has parallel sides” yes, “all sides equal” yes
    C: “has parallel sides” no, “all sides equal” no
    D: “has parallel sides” no, “all sides equal” yes
    E: “has parallel sides” yes, “all sides equal” no
    F: “has parallel sides” no, “all sides equal” no
  2. Put the shapes in the Venn diagram. Shapes that fit both labels go in the overlap.
    has parallel sidesall sides equalAEBDCF
  3. Write the answer.
    Only “has parallel sides”: A and E. Both: B. Only “all sides equal”: D. Neither: C and F.

Check. 2 + 1 + 1 + 2 = 6 shapes. ✓

Watch out. Shapes with neither property go outside the circles but inside the box.

Worked example 9.5.7
Which shape am I? “I have 4 sides. I have two pairs of parallel sides. I have no right angles. All my sides are the same length.”

Key words. A Venn diagram uses circles. Things that fit both labels go in the part where the circles overlap. A Carroll diagram is a table with yes and no.

Method. Use the clues one at a time. After each clue, cross out the shapes that cannot be the answer. The shape that is left is the answer.

  1. Use the clue: “I have 4 sides.” Cross out every shape that does not fit. Keep the ones that do.
    Keep: square, rectangle, rhombus, parallelogram, trapezium and kite.
    Cross out: equilateral triangle, isosceles triangle, scalene triangle, right-angled triangle, regular pentagon, regular hexagon, regular heptagon, regular octagon, regular nonagon and regular decagon.
  2. Use the clue: “I have two pairs of parallel sides.” Cross out every shape that does not fit. Keep the ones that do.
    Keep: square, rectangle, rhombus and parallelogram.
    Cross out: trapezium and kite.
  3. Use the clue: “I have no right angles.” Cross out every shape that does not fit. Keep the ones that do.
    Keep: rhombus and parallelogram.
    Cross out: square and rectangle.
  4. Use the clue: “All of my sides are the same length.” Cross out every shape that does not fit. Keep the ones that do.
    Keep: rhombus.
    Cross out: parallelogram.
  5. Write the answer. Only one shape fits every clue.
    The shape is the rhombus.

Check. Look at the rhombus: every clue is true for it. ✓

Watch out. Read every clue. One clue on its own often fits more than one shape.

Watch out

Equal sides are not enough for a regular polygon. A rhombus has four equal sides, but it is not regular because its angles are not all equal. A regular polygon needs equal sides and equal angles. Also, when you sort shapes, do not forget the shapes that fit neither label: they still go in the diagram.

Exercise 9E  Polygons

Non-calculator
1
FWrite the name of each polygon.
(a)
(b)
(c)
(d)
2
FHow many sides and how many vertices does each polygon have?
(a)hexagon
(b)octagon
(c)pentagon
(d)decagon
3
FWrite the name of the polygon that has…
(a)7 sides
(b)9 sides
(c)6 sides
(d)4 sides
4
FWhich of these shapes are polygons? Write the letters.
ABCDEF
5
FIs each polygon regular or irregular?
(a)
(b)
(c)
(d)
6
FFill in the gaps.
(a)A regular octagon has ___ equal sides.
(b)A regular hexagon has ___ vertices.
7
PSA road sign is a regular octagon. Each side is 25 cm long. What is the perimeter of the sign?
8
PSMei draws 5 separate hexagons. How many sides does she draw altogether?
9
RNoah says, “This hexagon has 6 equal sides, so it is a regular hexagon.” Is Noah correct? Explain.
10
RIs each statement true or false?
(a)A rectangle (that is not a square) is a regular polygon.
(b)A square is a regular polygon.
(c)A triangle is a polygon.
11
RA regular polygon has sides of 4 cm. Its perimeter is 32 cm. How many sides does it have? What is the polygon called?

Exercise 9F  Sorting shapes

Non-calculator
1
FSort shapes A to H into the Carroll diagram. Write the letters that go in each box.
ABCDEFGH
has a right angleno right angle
4 sides
not 4 sides
2
FSort shapes A to F into the Venn diagram. Write the letters in each part.
ABCDEF
has parallel sidesall sides equal
3
FThe Venn diagram shows shapes sorted by two labels. The right label is missing. Write a label for the right circle.
ABCDEF
has 4 sides?CDABEF
4
FHere is the Venn diagram for “has parallel sides” and “all sides equal”. Where does each new shape go? Write left circle only, right circle only, in the overlap or outside both circles.
has parallel sidesall sides equal
(a) rhombus
(b) kite
(c) rectangle
5
FWhich shape am I? Use the clues one at a time.
(a)I have 3 sides. All of my sides are the same length.
(b)I have 6 sides. I am a regular polygon.
(c)I have 4 sides. I have four lines of symmetry.
6
PSWhich shape am I? These riddles are harder.
(a)I have 4 sides. I have two pairs of equal sides, and the equal sides are next to each other.
(b)I have 4 sides. I have exactly one pair of parallel sides. I have no right angles.
(c)I have 4 sides. I have two pairs of parallel sides. Two of my angles are acute and two are obtuse. My sides are not all the same length. I have no right angles.
7
RWhich shape is the odd one out? Say why.
(a)
ABCD
(b)
ABCD
(c)
ABCD
8
RWrite always, sometimes or never for each sentence.
(a)A regular polygon has equal sides.
(b)A pentagon is a regular polygon.
(c)A hexagon has 7 sides.
(d)A quadrilateral has four right angles.
9
PSHow many of the shapes A to H (shown in the first question) are quadrilaterals?
ABCDEFGH
10
RZainab sorted shapes A to F into a Venn diagram, but she made one mistake. Which shape is in the wrong place? Where should it go?
ABCDEF
has parallel sidesall sides equalAEDCFB
11
RIn a Venn diagram, the left circle is “has 4 sides” and the right circle is “has a right angle”. Name two shapes that could go in the overlap. Explain how you know.
has 4 sideshas a right angle??
9.6

Lines of symmetry

Year 4
Lines of symmetry in 2-D shapes, Shapes in different orientations, Letters and patterns

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Lines of symmetryComing soon

A shape has a line of symmetry if you can fold it along a line so that the two halves fit exactly on top of each other. The line is also called a mirror line, because one half is the mirror image of the other half. Some shapes have one line, some have many, and some have none.

isosceles triangle: 1rectangle: 2square: 4parallelogram: 0
Red dashed lines are lines of symmetry.

Regular polygons have as many lines of symmetry as they have sides: a regular pentagon has 5 and a regular hexagon has 6.

3 sides: 3 lines4 sides: 4 lines5 sides: 5 lines6 sides: 6 lines
In a regular polygon the number of lines of symmetry is the same as the number of sides.

Different orientations. A shape can be turned round. A turned shape has the same number of lines of symmetry, but the lines point in different directions. A line of symmetry can be vertical (up and down), horizontal (across) or sloping.

vertical linehorizontal linesloping linesloping line
The same arrow shape in four positions. It always has exactly one line of symmetry.

Letters and patterns. Some capital letters have lines of symmetry. The letter H has two: one up and down and one across. The letter A has one. The letter N has none. Patterns made of coloured squares can also have lines of symmetry, even diagonal ones.

H: 2A: 1E: 1N: 0
a pattern with 4 lines
Letters H, A, E and N, with their lines of symmetry. The last picture is a square pattern with four lines.

Key ideas

  • A line of symmetry (mirror line) is a fold line that makes the two halves match exactly.
  • A shape can have 0, 1, 2 or more lines of symmetry. A regular polygon has as many lines as sides.
  • Turning a shape round does not change its number of lines of symmetry.
  • A rectangle has 2 lines of symmetry (not the diagonals). A parallelogram has none.
  • Check lines by folding, or by imagining a mirror. Letters and patterns can have lines too.
Worked example 9.6.1
How many lines of symmetry does this rectangle have?

Key words. A line of symmetry (mirror line) cuts a shape into two halves that are mirror images of each other. If you fold along the line, the two halves match exactly.

Method. Find the lines you could fold along so that the two halves match exactly. Count them. Turning the shape round does not change how many there are.

  1. Imagine folding the shape. A line of symmetry is a fold line where the two halves match exactly. The shape turned round still has the same lines of symmetry.
    The red dashed lines are the lines of symmetry.
  2. Look for every line. Try lines through the corners and through the middles of the sides.
    The rectangle has 2 lines of symmetry.
  3. Check the diagonals. A diagonal is a line from corner to corner. Fold along it: the corners do not fit on top of each other.
    The diagonals are not lines of symmetry.
  4. Write the answer.
    2

Check. Fold along each red line: the two halves fit exactly. ✓

Watch out. Turning a shape does not change how many lines of symmetry it has. The diagonals of a rectangle are not lines of symmetry.

Worked example 9.6.2
How many lines of symmetry does this isosceles triangle have?

Key words. A line of symmetry (mirror line) cuts a shape into two halves that are mirror images of each other. If you fold along the line, the two halves match exactly.

Method. Find the lines you could fold along so that the two halves match exactly. Count them. Turning the shape round does not change how many there are.

  1. Imagine folding the shape. A line of symmetry is a fold line where the two halves match exactly. The shape turned round still has the same lines of symmetry.
    The red dashed lines are the lines of symmetry.
  2. Look for every line. Try lines through the corners and through the middles of the sides.
    The isosceles triangle has 1 line of symmetry.
  3. Count the lines.
    The number of lines of symmetry is 1.
  4. Write the answer.
    1

Check. Fold along each red line: the two halves fit exactly. ✓

Watch out. Turning a shape does not change how many lines of symmetry it has. The diagonals of a rectangle are not lines of symmetry.

Worked example 9.6.3
How many lines of symmetry does this regular pentagon have?

Key words. A line of symmetry (mirror line) cuts a shape into two halves that are mirror images of each other. If you fold along the line, the two halves match exactly.

Method. Find the lines you could fold along so that the two halves match exactly. Count them. Turning the shape round does not change how many there are.

  1. Imagine folding the shape. A line of symmetry is a fold line where the two halves match exactly. The shape turned round still has the same lines of symmetry.
    The red dashed lines are the lines of symmetry.
  2. Look for every line. Try lines through the corners and through the middles of the sides.
    The regular pentagon has 5 lines of symmetry.
  3. Count the lines.
    The number of lines of symmetry is 5.
  4. Write the answer.
    5

Check. Fold along each red line: the two halves fit exactly. ✓

Watch out. Turning a shape does not change how many lines of symmetry it has. The diagonals of a rectangle are not lines of symmetry.

Worked example 9.6.4
Is the dashed line a line of symmetry of the rectangle? It goes from corner to corner.

Key words. A line of symmetry (mirror line) cuts a shape into two halves that are mirror images of each other. If you fold along the line, the two halves match exactly.

Method. Imagine folding the shape along the dashed line. If the two halves fit exactly on top of each other, it is a line of symmetry. The orange outline shows where the other half lands after the fold.

  1. Look at the dashed line. It is the fold line.
  2. Fold along the line. The orange outline is the reflection: where each corner lands after the fold.
  3. Compare the two halves. Do the orange lines lie exactly on the shape’s edges?
    No. The orange outline sticks out of the shape, so the halves do not fit.
  4. Answer.
    No, it is not a line of symmetry.

Check. A rectangle does not have a line of symmetry here. ✓

Watch out. A line that cuts a shape into two equal halves is not always a line of symmetry. The two halves must be mirror images.

Worked example 9.6.5
How many lines of symmetry does the letter E have?

Key words. A line of symmetry (mirror line) cuts a shape into two halves that are mirror images of each other. If you fold along the line, the two halves match exactly.

Method. Imagine a mirror standing up the middle of the letter (a vertical line) and another lying across the middle (a horizontal line). If one half is the mirror image of the other, it is a line of symmetry.

  1. Look at the letter E. The red dashed lines are the lines of symmetry.
  2. Test the up-and-down (vertical) line. Do the left and right halves match?
    No.
  3. Test the across (horizontal) line. Do the top and bottom halves match?
    Yes.
  4. Count.
    The letter E has 1 line of symmetry.

Check. Fold a copy of the letter along each line to check. ✓

Watch out. Look at the letter as a block shape.

Worked example 9.6.6
How many lines of symmetry does this pattern of shaded squares have?

Key words. A line of symmetry (mirror line) cuts a shape into two halves that are mirror images of each other. If you fold along the line, the two halves match exactly.

Method. Imagine a mirror along each possible line: up and down, across, and the two corner-to-corner lines. Check each one: does the shaded pattern match on both sides?

  1. Test the vertical line (up and down). Compare the left and right sides.
    Does not match.
  2. Test the horizontal line (across). Compare the top and bottom.
    Does not match.
  3. Test the two diagonal lines. Compare the two sides of each corner-to-corner line.
    The \ diagonal matches. The / diagonal does not match.
  4. Count the lines. The red lines are the lines of symmetry.
    The pattern has 1 line of symmetry.

Check. Fold along each red line: the shaded squares land on shaded squares. ✓

Watch out. Do not forget the diagonal lines.

Worked example 9.6.7
How many lines of symmetry does this parallelogram have?

Key words. A line of symmetry (mirror line) cuts a shape into two halves that are mirror images of each other. If you fold along the line, the two halves match exactly.

Method. Find the lines you could fold along so that the two halves match exactly. Count them. Turning the shape round does not change how many there are.

  1. Imagine folding the shape. A line of symmetry is a fold line where the two halves match exactly. The shape turned round still has the same lines of symmetry.
    There is no fold line that makes the two halves match.
  2. Look for every line. Try lines through the corners and through the middles of the sides.
    The parallelogram has no lines of symmetry.
  3. Try some fold lines. Fold along a diagonal or along a line through the middle. The halves do not match.
    No fold line works.
  4. Write the answer.
    0

Check. Try folding in several ways: the halves never fit. ✓

Watch out. Turning a shape does not change how many lines of symmetry it has. The diagonals of a rectangle are not lines of symmetry.

Watch out

The diagonals of a rectangle are not lines of symmetry. A rectangle has only 2 lines: through the middles of opposite sides. If you fold it along a diagonal, the corners do not land on each other. Also remember: a line that cuts a shape into two equal pieces is not always a line of symmetry. The two halves must be mirror images.

Exercise 9G  Lines of symmetry

Non-calculator
1
FIs the dashed line a line of symmetry? Write yes or no.
(a)
(b)
(c)
(d)
2
FHow many lines of symmetry does each shape have?
(a)
(b)
(c)
(d)
(e)
(f)
3
FWhich shapes have no lines of symmetry? Write the letters.
ABCDEF
4
FWhich shape has exactly one line of symmetry? Write the letter.
ABCD
5
FHow many lines of symmetry does each letter have?
(a)H
(b)A
(c)Z
(d)C
6
FLook at these capital letters. Write the letters that have…
ADEHMNTVWZ
(a)a vertical line of symmetry (up and down)
(b)a horizontal line of symmetry (across)
7
FHow many lines of symmetry does each pattern have? Remember the diagonal lines.
(a)
(b)
(c)
(d)
8
FHow many lines of symmetry does each regular polygon have?
(a)a regular heptagon
(b)a regular octagon
(c)a regular decagon
9
FEach shape has one line of symmetry. Is the line vertical, horizontal or sloping?
(a)
(b)
(c)
10
PSRavi folds a square piece of paper so that the two halves match exactly. How many different folds can he make?
11
RAli says, “A rectangle has 4 lines of symmetry: 2 diagonals and 2 middle lines.” Do you agree? Explain.
12
RWrite always, sometimes or never for each sentence.
(a)A shape with four equal sides has four lines of symmetry.
(b)A regular polygon has the same number of lines of symmetry as sides.
13
RWhich shape am I? “I have 4 sides. My sides are not all the same length. I have two lines of symmetry.”
9.7

Completing symmetrical figures

Year 4
Reflection in a mirror line, Complete a symmetric figure, Vertical, horizontal and diagonal mirror lines

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Completing symmetrical figuresComing soon

A symmetrical figure looks the same on both sides of its mirror line. If you are given one half and the mirror line, you can complete the figure by drawing the other half. Each point in the new half is the same distance from the mirror line as the matching point in the first half, but on the other side.

  1. Find the mirror line.
  2. Take one shaded square (or one corner). Count how far it is from the mirror line.
  3. Go the same distance on the other side and shade the matching square.
  4. Do every square. Check that both sides match.
ABCDEFGH123456half givenABCDEFGH123456completed
A vertical mirror line. Orange squares were given and green squares were drawn to complete the figure.

Different mirror lines. The mirror line can be vertical (up and down), horizontal (across) or diagonal (corner to corner). For a diagonal line, a square swaps its row and column with its partner: the square in column D, row 2 goes to column B, row 4.

ABCDEF12345678horizontal mirror lineABCDEF123456diagonal mirror line
The completed figures. Green squares are the reflection.

Key ideas

  • A symmetrical figure matches on both sides of the mirror line.
  • Each point and its partner are the same distance from the mirror line, on opposite sides.
  • Count squares (or dots) from the mirror line, not from the edge of the grid.
  • For a diagonal mirror line, the row and the column swap.
  • Check by folding: the two sides must match exactly.
Worked example 9.7.1
Complete the shaded figure so that it is symmetrical about the vertical mirror line.
ABCDEFGH123456

Key words. A mirror line is the line a shape is reflected in. Each point of the reflection is the same distance from the mirror line as the point it matches, but on the other side.

Method. Reflect each shaded square in the mirror line. Each square and its partner are the same distance from the mirror line, on opposite sides.

  1. Find the mirror line. The red dashed line is the mirror line.
    The mirror line is a vertical line (up and down).
  2. Find the partner of each shaded square. Count the squares between each shaded square and the mirror line. The partner has the same number of squares between it and the mirror line, on the other side.
    A4 has 3 squares between it and the mirror line, so its partner has 3 squares between it and the mirror line on the other side: H4.
    B3 has 2 squares between it and the mirror line, so its partner has 2 squares between it and the mirror line on the other side: G3.
    B4 has 2 squares between it and the mirror line, so its partner has 2 squares between it and the mirror line on the other side: G4.
    C2 has 1 square between it and the mirror line, so its partner has 1 square between it and the mirror line on the other side: F2.
    C3 has 1 square between it and the mirror line, so its partner has 1 square between it and the mirror line on the other side: F3.
    C4 has 1 square between it and the mirror line, so its partner has 1 square between it and the mirror line on the other side: F4.
    D1 touches the mirror line, so E1 touches it on the other side.
    D2 touches the mirror line, so E2 touches it on the other side.
    D3 touches the mirror line, so E3 touches it on the other side.
    D4 touches the mirror line, so E4 touches it on the other side.
    D5 touches the mirror line, so E5 touches it on the other side.
    D6 touches the mirror line, so E6 touches it on the other side.
  3. Shade the new squares. Orange squares were given. Green squares are the new ones.
    ABCDEFGH123456
    Shade 12 more squares: E1, E2, E3, E4, E5, E6, F2, F3, F4, G3, G4, H4.

Check. Fold along the mirror line: every green square lands on an orange square. ✓

Watch out. Count the squares from the mirror line, not from the edge of the grid.

Worked example 9.7.2
Complete the shaded figure so that it is symmetrical about the horizontal mirror line.
ABCDEF12345678

Key words. A mirror line is the line a shape is reflected in. Each point of the reflection is the same distance from the mirror line as the point it matches, but on the other side.

Method. Reflect each shaded square in the mirror line. Each square and its partner are the same distance from the mirror line, on opposite sides.

  1. Find the mirror line. The red dashed line is the mirror line.
    The mirror line is a horizontal line (across).
  2. Find the partner of each shaded square. Each square goes the same distance from the mirror line, on the other side (up and down).
    A4 touches the mirror line, so A5 touches it on the other side.
    B3 has 1 square between it and the mirror line, so its partner has 1 square between it and the mirror line on the other side: B6.
    B4 touches the mirror line, so B5 touches it on the other side.
    C2 has 2 squares between it and the mirror line, so its partner has 2 squares between it and the mirror line on the other side: C7.
    C3 has 1 square between it and the mirror line, so its partner has 1 square between it and the mirror line on the other side: C6.
    C4 touches the mirror line, so C5 touches it on the other side.
    D2 has 2 squares between it and the mirror line, so its partner has 2 squares between it and the mirror line on the other side: D7.
    D3 has 1 square between it and the mirror line, so its partner has 1 square between it and the mirror line on the other side: D6.
    D4 touches the mirror line, so D5 touches it on the other side.
    E3 has 1 square between it and the mirror line, so its partner has 1 square between it and the mirror line on the other side: E6.
    E4 touches the mirror line, so E5 touches it on the other side.
    F4 touches the mirror line, so F5 touches it on the other side.
  3. Shade the new squares. Orange squares were given. Green squares are the new ones.
    ABCDEF12345678
    Shade 12 more squares: A5, B5, B6, C5, C6, C7, D5, D6, D7, E5, E6, F5.

Check. Fold along the mirror line: every green square lands on an orange square. ✓

Watch out. Count the squares from the mirror line, not from the edge of the grid.

Worked example 9.7.3
Complete the shaded figure so that it is symmetrical about the diagonal mirror line.
ABCDEF123456

Key words. A mirror line is the line a shape is reflected in. Each point of the reflection is the same distance from the mirror line as the point it matches, but on the other side.

Method. Reflect each shaded square in the mirror line. Each square and its partner are the same distance from the mirror line, on opposite sides.

  1. Find the mirror line. The red dashed line is the mirror line.
    The mirror line is a diagonal line (from the top left corner to the bottom right corner).
  2. Find the partner of each shaded square. With a diagonal mirror line, a square in column C and row R goes to column R and row C. The row and column swap.
    A1 is on the mirror line, so it stays where it is.
    A2: swap the column and row, so its partner is B1.
    B2 is on the mirror line, so it stays where it is.
    B3: swap the column and row, so its partner is C2.
    B4: swap the column and row, so its partner is D2.
    C4: swap the column and row, so its partner is D3.
    C5: swap the column and row, so its partner is E3.
    C6: swap the column and row, so its partner is F3.
    D6: swap the column and row, so its partner is F4.
  3. Shade the new squares. Orange squares were given. Green squares are the new ones.
    ABCDEF123456
    Shade 7 more squares: B1, C2, D2, D3, E3, F3, F4.

Check. Fold along the mirror line: every green square lands on an orange square. ✓

Watch out. Count the squares from the mirror line, not from the edge of the grid.

Worked example 9.7.4
Complete the shape so that it is symmetrical about the mirror line. Corners A, B and C are the corners that are not on the mirror line.
ABC

Key words. A mirror line is the line a shape is reflected in. Each point of the reflection is the same distance from the mirror line as the point it matches, but on the other side.

Method. Reflect each corner (vertex) of the shape in the mirror line. Corners on the mirror line stay where they are. Then join the new corners to make the other half of the shape.

  1. Find the mirror line. The red dashed line is the mirror line. Some corners of the shape are on it.
    The mirror line is a vertical line (up and down).
  2. Find the partner of each corner that is not on the line. Each partner is the same distance from the mirror line, on the other side.
    Corner A is 3 dots to the right of the mirror line and 0 dots up. Its partner is 3 dots to the left of the line, 0 up.
    Corner B is 3 dots to the right of the mirror line and 3 dots up. Its partner is 3 dots to the left of the line, 3 up.
    Corner C is 2 dots to the right of the mirror line and 5 dots up. Its partner is 2 dots to the left of the line, 5 up.
  3. Join the corners. Join the new corners in order to finish the shape. Orange is the half you were given and green is the new half.
    ABC

Check. Fold along the mirror line: the two halves match exactly. The completed shape is a hexagon. ✓

Watch out. Take care to count the dots from the mirror line, not from the edge of the grid.

Worked example 9.7.5
Complete the shape so that it is symmetrical about the diagonal mirror line.
AB

Key words. A mirror line is the line a shape is reflected in. Each point of the reflection is the same distance from the mirror line as the point it matches, but on the other side.

Method. Reflect each corner (vertex) of the shape in the mirror line. Corners on the mirror line stay where they are. Then join the new corners to make the other half of the shape.

  1. Find the mirror line. The red dashed line is the mirror line. Some corners of the shape are on it.
    The mirror line is a diagonal line (from the top left corner to the bottom right corner).
  2. Find the partner of each corner that is not on the line. With a diagonal mirror line, the numbers across and up swap.
    Corner A is 4 across and 1 up. Swap the numbers: its partner is 1 across and 4 up.
    Corner B is 5 across and 3 up. Swap the numbers: its partner is 3 across and 5 up.
  3. Join the corners. Join the new corners in order to finish the shape. Orange is the half you were given and green is the new half.
    AB

Check. Fold along the mirror line: the two halves match exactly. The completed shape is a hexagon. ✓

Watch out. Take care to count the dots from the mirror line, not from the edge of the grid.

Worked example 9.7.6
Zainab completed the figure, but she made one mistake. Which square is wrong? Where should it be?
ABCDEFGH123456

Key words. A mirror line is the line a shape is reflected in. Each point of the reflection is the same distance from the mirror line as the point it matches, but on the other side.

Method. Check each shaded square against its partner. Find the square in the wrong place.

  1. Look at the attempt. Orange squares are shaded. The mirror line is vertical.
    ABCDEFGH123456
  2. Reflect the original squares. Work out where every partner should be.
    The new squares should be: E1, E2, E3, E4, E5, E6, F2, F3, F4, G3, G4, H4.
  3. Compare. Find a shaded square that should not be there, and a square that is missing.
    is shaded but should not be. F2 is missing.
  4. Show the corrected figure. The wrong square is red. The correct square is green.
    ABCDEFGH123456

Check. Every square on the right is now the mirror image of a square on the left. ✓

Watch out. Check the figure square by square.

Watch out

Count from the mirror line, not from the edge. A square that is 2 squares from the mirror line has a partner that is 2 squares from the mirror line on the other side. Also, do not slide the shape across: a reflection is a flip, so the shape faces the other way.

Exercise 9H  Complete the symmetrical figure

Non-calculator
1
FShade squares to complete each figure. The mirror line is vertical. Write the squares you shade.
(a)ABCDEFGH123456
(b)ABCDEFGH123456
2
FShade squares to complete each figure. The mirror line is horizontal.
(a)ABCDEF12345678
(b)ABCDEF12345678
3
FThe mirror line is a diagonal line. Shade squares to complete each figure.
(a)ABCDEF123456
(b)ABCDEF123456
4
FComplete the shape so that it is symmetrical about the mirror line. Say what the completed shape is.
(a)ABCD
(b)ABC
5
FComplete the shape so that it is symmetrical about the diagonal mirror line. Say what the completed shape is.
AB
6
FHow many squares will be shaded when the figure is complete? The mirror line is vertical.
ABCDEFGH123456
7
FWhich picture shows the shape correctly reflected in the mirror line? Write the letter.
ABC
8
PSZainab completed this figure. She made one mistake. Which square is wrong, and where should it go?
ABCDEFGH123456
9
PSThis pattern should be symmetrical about the mirror line, but some squares have been left out. Which squares must be shaded? Write them.
ABCDEFGH123456
10
RThis pattern has two mirror lines. Complete it so that it is symmetrical about both lines. How many squares are shaded in the finished pattern?
ABCDEF123456
11
RThe grid shows 6 shaded squares and a diagonal mirror line. Mia says, “When the figure is complete there will be 12 shaded squares, because 6 + 6 = 12.” Is Mia right? Explain.
ABCDEF123456
12
RThe mirror line is the diagonal line from the top left corner to the bottom right corner of a square grid. Is each person right?
ABCDEF123456
(a)Priya says the partner of D2 is B4.
(b)Noah says the partner of C4 is C3.

Chapter summary

SkillMethod
TurnsWhole turn = 4 right angles. Half turn = 2 (a straight line). Quarter turn = 1. Three-quarter turn = 3. Clockwise goes the way a clock goes.
AnglesAcute: smaller than a right angle. Right angle: a quarter turn. Obtuse: bigger than a right angle, smaller than a straight line. Compare with a paper corner. Arm length does not matter.
Perpendicular and parallelPerpendicular lines meet at a right angle. Parallel lines never meet and stay the same distance apart.
TrianglesEquilateral: 3 equal sides. Isosceles: 2 equal sides. Scalene: none. Right-angled: one right angle. Tick marks show equal sides.
QuadrilateralsSquare, rectangle, rhombus, parallelogram (2 pairs of parallel sides), trapezium (exactly 1 pair), kite (2 pairs of equal sides next to each other). A square is a special rectangle.
PolygonsPentagon 5, hexagon 6, heptagon 7, octagon 8, nonagon 9, decagon 10 sides. Regular: all sides and all angles equal. Sides = vertices.
SortingCarroll diagram: a table with yes and no. Venn diagram: both labels go in the overlap, neither goes outside the circles.
Lines of symmetryFold line where both halves match. Regular polygon: as many lines as sides. Rectangle 2, square 4, parallelogram 0. Turning does not change the number.
Complete a symmetric figureFind each point’s partner: same distance from the mirror line, other side. Diagonal mirror line: row and column swap.

Can you…?

  • I can say how many right angles are in a quarter, half, three-quarter and whole turn.
  • I can name acute, right and obtuse angles and a straight line.
  • I can compare and order angles by size, even when the arms have different lengths.
  • I can name equilateral, isosceles, scalene and right-angled triangles.
  • I can name the six quadrilaterals and find parallel and perpendicular sides.
  • I can name polygons with 5 to 10 sides and say if they are regular.
  • I can sort shapes in Venn and Carroll diagrams and solve “Which shape am I?” riddles.
  • I can find the lines of symmetry of shapes, letters and patterns, however they are turned.
  • I can complete a symmetrical figure using a vertical, horizontal or diagonal mirror line.

Key terms

Turn
A movement round a point. Whole, half, quarter and three-quarter turns.
Right angle
A quarter turn. The corner of a book. Marked with a small square.
Acute angle
An angle smaller than a right angle.
Obtuse angle
An angle bigger than a right angle but smaller than a straight line.
Straight line (angle)
Two right angles, a half turn.
Clockwise / anticlockwise
The way a clock’s hands go / the opposite way.
Perpendicular
Meeting at a right angle.
Parallel
Lines that stay the same distance apart and never meet.
Equilateral triangle
A triangle with 3 equal sides.
Isosceles triangle
A triangle with 2 equal sides.
Scalene triangle
A triangle with no equal sides.
Right-angled triangle
A triangle with one right angle.
Quadrilateral
A shape with 4 straight sides.
Rhombus
A quadrilateral with 4 equal sides and no right angles.
Parallelogram
A quadrilateral with two pairs of parallel sides.
Trapezium
A quadrilateral with exactly one pair of parallel sides.
Kite
A quadrilateral with two pairs of equal sides next to each other.
Polygon
A closed flat shape with straight sides.
Vertex
A corner of a shape (plural: vertices).
Regular polygon
A polygon with all sides equal and all angles equal.
Irregular polygon
A polygon that is not regular.
Venn diagram
A sorting diagram made of circles.
Carroll diagram
A sorting diagram that is a table with yes and no.
Line of symmetry
A fold line where the two halves of a shape match exactly.
Mirror line
The line a shape is reflected in.
Reflection
A flip of a shape in a mirror line.

Exercise 9Z  Check-up (non-calculator)

Non-calculator
1
Mei is facing South. She makes a quarter turn clockwise. Which way is she facing?
NESW
2
Is this angle acute, a right angle or obtuse?
3
Write the angles in order, smallest first.
ABC
4
Name this triangle. The sides are in centimetres.
6 cm6 cm4 cm
5
What is the name of this quadrilateral?
6
Write the full name of this polygon. All sides are equal and all angles are equal.
7
Which shape am I? “I have 4 sides. I have two pairs of equal sides, and the equal sides are next to each other.”
8
How many lines of symmetry does this triangle have?
9
Complete the figure so that it is symmetrical about the mirror line. Write the squares you shade.
ABCDEFGH123456
10
Write always, sometimes or never: a square is a rectangle.

Total 18 marks. Allow about 18 minutes.

Exercise 9X  SATs-style questions

Non-calculator
1
FWhich angle is obtuse? Write the letter.[1 mark]
ABCD
2
FTick the two quadrilaterals. Write the letters.[1 mark]
ABCDE
3
FWhat is the name of this triangle? The sides are in centimetres.[1 mark]
9 cm7 cm6 cm
4
FHow many lines of symmetry does this shape have?[1 mark]
5
FComplete the pattern so that it is symmetrical about the mirror line. Write the squares you shade.[2 marks]
ABCDEF12345678
6
PSWrite the letters of shapes A to F in the correct boxes of the Carroll diagram.[2 marks]
ABCDEF
has parallel sidesnot: has parallel sides
all sides equal
not: all sides equal
7
RWhich shape am I? “I have 4 sides. I have two pairs of parallel sides. I have four right angles. My sides are not all the same length.” Show how you know.[2 marks]
8
RBen says, “All rectangles are squares.” Is he right? Explain.[2 marks]
9
FWrite the angles in order, smallest first. Then write which angle is the right angle.[3 marks]
ABCD
10
PSA robot faces East. It makes a half turn clockwise, then a quarter turn anticlockwise, then a three-quarter turn clockwise. (a) Which way is the robot facing now? (b) How many right angles has the robot turned through altogether?[3 marks]
NESW
← Chapter 8: TimeChapter 10: Position and direction →

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